Verify that f has an inverse and find (f^-1)^'(c)....

Key Concepts

Invertibility of Functions

A function must be one-to-one (injective) on its domain to have an inverse. This property means that for every output there is exactly one input. In the context of the function given, showing that tan x is strictly increasing on the interval (??/2, ?/2) confirms that it is injective and thus invertible.

Inverse Function Theorem

This theorem provides a direct method to compute the derivative of an inverse function. Specifically, if f is differentiable and invertible at a point, then the derivative of its inverse at a corresponding value is the reciprocal of the derivative of the function evaluated at the inverse image of that value. This is expressed as (f?¹)'(y) = 1 / f'(f?¹(y)).

Differentiation of Trigonometric Functions

Understanding the derivative of the tangent function is essential in this context. The derivative of tan x is sec²x, which plays a critical role when applying the inverse function theorem. Recognizing and computing the derivative of trigonometric functions helps in evaluating the derivative of the inverse function at a specific point.

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